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The Standard Model is a theory describing the elementary particle interactions. The electromagnetic, weak and strong interactions have been involved in the theory exploiting the relation between group symmetries and group generators, in order to wisely define the equation describing both of the leptons and bosons wave equation. These particles wave equation lead to the formulation of the Standard Model Lagrangian and also help for understanding well the gauge theory. The most successful in the theory was its prediction of the Higgs Field, which together with the spontaneous symmetry breaking, gives mass to the vector bosons and to the fermions and quarks. The discovery of neutral currents which was accomplished by CERN neutrino experiment in 1973, the discovery of charm quark in 1974, the detection of the existence of the ????± and ????0 in 1983 in CERN, The discovery of the top quark in 1994 by the Tevatron experiments, and the existence of the Higgs boson which was discovered by the LHC experiments in 2012 revealed the accuracy of the theory. Even though this theory is one of the greatest achievements of particle physics of the twentieth century, it is incomplete because it does not describe the physics of the dark matter, the baryon asymmetry, the neutrino oscillations etc. That’s why we need to extend the theory in which we focus on the Grassmannian Model with which we may discover the mister of the dark matter.